Implication Graphs and Logic Testing

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1 Implition Grphs n Logi Testing Vishwni D. Agrwl Jmes J. Dnher Professor Dept. of ECE, Auurn University Auurn, AL vgrwl@eng.uurn.eu Joint reserh with: K. K. Dve, ATI Reserh, Yrley, PA M. L. Bushnell, Rutgers University, Pistwy,NJ 2/2/05 VLSI D&T Group Seminr

2 Implition Grph An implition grph (IG) represents the implition reltions etween pirs of Boolen vriles. n implition ontrpositive implition 2/2/05 VLSI D&T Group Seminr 2

3 Implition Grph of Logi Gte = Boolen flse funtion: = = 0 Chkrhr et l. -- IEEE-D&T, 990 2/2/05 VLSI D&T Group Seminr 3

4 Glol Implitions n Trnsitive Closure Trnsitive losure ege 0 2/2/05 VLSI D&T Group Seminr 4

5 Trnsitive Closure Trnsitive losure (TC) of irete grph ontins the sme set of noes s the originl grph. If there is irete pth from noe to, then the trnsitive losure ontins n ege from to. A Grph Trnsitive Closure A grph /2/05 VLSI D&T Group Seminr

6 Trnsitive Closure: Wrshll s Algorithm proeure Wrshll (vr A : rry[ n, n] of oolen; C : rry[ n, n] of oolen); { Wrshll mkes A the trnsitive losure of C } vr i, j, k : integer; egin for i := to n o for j := to n o A[i, j] := C[i, j]; for k := to n o O(n 3 ) for i := to n o for j := to n o if A[i, j] = flse then A[i, j] := A[i, k] n A[k, j] en; { Wrshll } S. Wrshll, A Theorem on Boolen Mtries, J. ACM, vol. 9, no., pp. -2. A. V. Aho, J. E. Hoproft n J. D. Ullmn, Dt Strutures n Algorithms, Reing, Msshusetts: Aison-Wesley, 983, p /2/05 VLSI D&T Group Seminr 6

7 Trnsitive Closure: Upte Algorithm Strt onstruting trnsitive losure (TC) y pling ll noes n no eges. This ege-less grph is its own TC. A eges to TC in ny ritrry orer For eh ege i j fin P : set of prent noes of i C : set of hil noes of j A eges {P, i } {C, j } K. Dve, Using Contrpositive Rule to Enhne the Implition Grphs of Logi Ciruits, Mster s Thesis, Rutgers University, Dept. of ECE, Pistwy, New Jersey, My /2/05 VLSI D&T Group Seminr 7

8 Upte Algorithm p i p2 {P, i } {C, j } j 2/2/05 VLSI D&T Group Seminr 8

9 Upte Algorithm Exmple A irete grph Trnsitive losure 2/2/05 VLSI D&T Group Seminr 9

10 Logi Testing: Stuk-t Fult A type of fult, whih uses line to hol onstnt logi vlue, irrespetive of hnge of stte t previous stges. There re two types of stuk-t-fults: Stuk-t- Stuk-t-0 Detetion of fult requires the fult to e tivte n its effet oserve t primry output (PO). Fult s-- is etetle, if following onitions re simultneously stisfie: = 0 O = fult is tivte oservility is true 2/2/05 VLSI D&T Group Seminr 0

11 Oservility Vriles Oservility vrile of signl represents whether or not tht signl is oservle t PO. It n e true or flse. O O = O (PO) O O C O = 0 O O + O + O O = 0 Agrwl, Bushnell n Lin, Reunny Ientifition using Trnsitive Closure, Pro. Asin Test Symp., 996, pp O 2/2/05 VLSI D&T Group Seminr

12 Reunnt Fults A fult tht hs no test is lle n untestle fult. Any untestle fult in omintionl iruit is reunnt fult euse it oes not use ny hnge in the input/output logi funtion of the iruit. Ientifition of reunnt fults is useful euse they n e remove from testing onsiertion, or from hrwre Fult stuk-t- is reunnt if either or O or O or O no ontrollility no oservility no rivility no rivility 2/2/05 VLSI D&T Group Seminr 2

13 Limittion of Implition Grph s--0 e s--0 Ciruit with two reunnt fults O O Implition grph (some noes n eges not shown) Implition grph shows no implitions of n on their oservilities. 2/2/05 VLSI D&T Group Seminr 3

14 Aing Prtil Implitions = Boolen flse funtion: = = 0 Λ V Henftling n Wittmnn, AEÜ, 995 (Λ noe) Dve, Mster s Thesis, 2004 (V noe) Λ n V noes represent prtil implitions 2/2/05 VLSI D&T Group Seminr 4

15 Using Prtil Implitions s--0 e s--0 Ciruit with two reunnt fults O O Implition grph (some noes n eges not shown) Implition Prtil implition Trnsitive losure ege 2/2/05 VLSI D&T Group Seminr 5

16 Another Exmple s--0 s-- s--0 s-- s--0 e e Λ 2 Λ Λ 3 e V V 2 Λ 4 2/2/05 VLSI D&T Group Seminr 6

17 Results on ISCAS Ciruits Reunnt fults ientifie n run time Ciruit Totl fults TRAN Chkrhr et l. FIRE Iyer n Armovii Imp. grph Meht et l. Enhne Imp. Grph Dve et l. Re. fults CPU s Re. Fults CPU s Re. Fults CPU s Re. Fults CPU s s Sun SPARC5 CPU Se. Sun SPARC2 CPU Se. 2/2/05 VLSI D&T Group Seminr 7

18 Referene Methos TRAN ATPG S. T. Chkrhr, V. D. Agrwl n S. G. Rothweiler, A Trnsitive Closure Algorithm for Test Genertion, IEEE Trns. CAD, vol. 2, no. 7, pp , July 993. FIRE Implition nlysis M. A. Iyer n M. Armovii, FIRE: A Fult-Inepenent Comintionl Reunny Ientifition Algorithm, IEEE Trns. VLSI Systems, vol. 4, no. 2, pp , June 996. Implition Grph V. J. Meht, Reunny Ientifition in Logi Ciruits using Extene Implition Grph n Stem Unoservility Theorems, Mster s Thesis, Rutgers University, Dept. of ECE, New Brunswik, NJ, My K. K. Dve, Using Contrpositive Rule to Enhne the Implition Grphs of Logi Ciruits, Mster s Thesis, Rutgers University, Dept. of ECE, New Brunswik, NJ, My /2/05 VLSI D&T Group Seminr 8

19 C908: Unientifie Reunnies Reunnt fults (s--) / Totl reunnt fults = 7; ientifie = 5 2/2/05 VLSI D&T Group Seminr 9

20 C535: Unientifie Reunny Reunnt fult (s--) PI 0/ PI 0 0/ 0 0/ 0 0/ 0 0 PO Totl reunnt fults = 59; ientifie = 58 2/2/05 VLSI D&T Group Seminr 20

21 0 C535: Continue... Reunnt fult (s--) PI 0/ PI 0/ 0/ 0 0 0/ PO 0 0 2/2/05 VLSI D&T Group Seminr 2

22 Conlusion Prtil implitions improve fult-inepenent reunny ientifition present results re the est known. Trnsitive losure omputtion run times re empirilly liner in the numer of noes for enhmrk iruits -- the known worst-se omplexity is O(N 3 ) for N noes. Upte lgorithm n effiiently ompute trnsitive losure when implition grph hs sprse onnetivity. Wekness of implition metho: Oservility of fnout stems. Reent work hs shown tht mny unoservle fnout stems n e ientifie from trnsitive losure nlysis. Reonvergent gte Domintor 2/2/05 VLSI D&T Group Seminr 22 Oservility of hs no iret reltion to oservilities of n, ut n e relte to tht of

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